Question: Find the value of $x$ that satisfies $\frac{\sqrt{5x}}{\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form.
We begin by multiplying out the denominator and then  squaring both sides \begin{align*}
\frac{\sqrt{5x}}{\sqrt{3(x-1)}}&=2\\
(\sqrt{5x})^2 &=\left(2\sqrt{3(x-1)}\right)^2\\
5x &= 12(x-1)\\
12& =7x\\
x&=\boxed{\frac{12}{7}}.\\
\end{align*}Checking, we see that this value of $x$ satisfies the original equation, so it is not an extraneous solution.